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Chapter 2: Problem 53

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=x^{4}-5 x^{2}-2 x$$

Short Answer

Expert verified
The zeros of the polynomial \(P(x)=x^{4}-5x^{2}-2x\) are \(x = -1.6\), \(x = 0\), \(x = 0.31\), and \(x = 3.29\). All zeros are distinct and thus each has a multiplicity of one.

Step by step solution

01

Identify the Polynomial

Identify the polynomial function \(P(x)=x^{4}-5x^{2}-2x\).
02

Simplify the polynomial

Factor out the common term, which is \(x\), from the expression. The simplified polynomial can be written as \(P(x)=x(x^{3}-5x-2)\).
03

Find the zeros

Set the polynomial equal to zero and solve for \(x\). From \(P(x) = x(x^{3}-5x-2)\), we know that \(x=0\) is a zero of the polynomial.
04

Solve the cubic polynomial

Next, solve \(x^{3}-5x-2=0\). To find the solutions for this expression, we can apply either the synthetic division method or solve using a suitable method in which you feel comfortable. The solutions are approximately \(x = -1.6 \), \(x = 0.31\), and \(x = 3.29 \).
05

State the multiplicities

All zeros of the polynomial are distinct, therefore all of them have a multiplicity of one.

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Most popular questions from this chapter

PIECES AND CUTS One straight cut through a thick piece of cheese produces two pieces. Two straight cuts can produce a maximum of four pieces. Three straight cuts can produce a maximum of eight pieces. You might be inclined to think that every additional cut doubles the previous number of pieces. However, for four straight cuts, you get a maximum of 15 pieces. The maximum number of pieces \(P\) that can be produced by \(n\) straight cuts is given by $$P(n)=\frac{n^{3}+5 n+6}{6}$$ a. Use the above function to determine the maximum number of pieces that can be produced by five straight cuts. b. What is the fewest number of straight cuts that are needed to produce 64 pieces?

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4$$

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=3 x^{6}-10 x^{5}-29 x^{4}+34 x^{3}+50 x^{2}-24 x-24$$

Show that if \(x=1+2 i,\) then \(x^{2}-2 x+5=0\)

Find the zeros of each polynomial function. If a zero is a multiple zero, state its multiplicity. $$P(x)=2 x^{3}+x^{2}-25 x+12$$
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